I will add an answer on the question about *homeomorphisms*.

As I said in my comments before, by using Seiberg-Witten theory one proves that, given *any* homeomorphism $\phi \colon X \to X'$ between two smooth $4$-manifolds, one has either $\phi(K_X)=K_{X'}$ or $\phi(K_X)=-K_{X'}$.

The second case *may occur*, even if $X$ and $X'$ are algebraic surfaces. In fact, in his paper [Orientation reversing homeomorphisms in surface geography, Math. Ann. 292 (1992)], D. Kotschick proved the following result:

>**Theorem.** 
There exist infinitely many pairs of simply connected algebraic surfaces of general type which are orientation-reversing homeomorphic, but not diffeomorphic.

He also makes a conjecture about *orientation-reversing diffeomorphic* algebraic surfaces:

>**Conjecture.** If two algebraic surface with finite fundamental group are orientation-reversing diffeomorphic, then they are homeomorphic to a geometrically ruled rational surface. In particular, they are simply connected.

I do not know the current state of this conjecture.